# What are the critical values, if any, of #f(x)=(4x)/(x^2 -1)#?

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The critical values of the function ( f(x) = \frac{4x}{x^2 - 1} ) occur where the derivative of the function is equal to zero or undefined. To find the critical values, we first need to find the derivative of the function ( f(x) ), then solve for ( x ) when the derivative is zero or undefined.

Taking the derivative of ( f(x) ) with respect to ( x ) using the quotient rule:

[ f'(x) = \frac{(4)(x^2 - 1) - (4x)(2x)}{(x^2 - 1)^2} ]

Simplifying the expression:

[ f'(x) = \frac{4x^2 - 4 - 8x^2}{(x^2 - 1)^2} ] [ f'(x) = \frac{-4x^2 - 4}{(x^2 - 1)^2} ]

To find critical values, set the derivative equal to zero and solve for ( x ):

[ -4x^2 - 4 = 0 ] [ -4x^2 = 4 ] [ x^2 = -1 ]

This equation has no real solutions, which means there are no critical values for this function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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